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Robust decentralized control in power systems
Claudio De Persis
Institute of Engineering and TechnologyJ.C. Willems Center for Systems and Control
PowerWeb Lunch LectureTU Delft, December 13, 2018
Joint work with Weitenberg (RUG), Jiang-Mallada (JohnsHopkins), Zhao (NREL), Dorfler (ETH)
AC power systems
• Power system = networkof generation, loads,transmission lines
• Power system control =maintain system securityat minimal cost
• Basic security requirement= keeping frequencyaround nominal value
1 / 22
Frequency controlFrequency control: Basics
• Any instantaneous load-generation imbalance results in afrequency deviation from the nominal one (50-60 Hz)
• Small load changes on a fast time scale are dealt with theAutomatic Generation Control (AGC)
2 / 22
The three control layers
Fig.�6.� Inertial,�primary�frequency�controls,�and�AGC�(secondary)�response�(figure�courtesy�Pouyan�Pourbeik�of�EPRI).�
by� individual� power� plants� adjusting� their� power� levels� in�response� to� requests� from�the�systems�operator.�
B.� Motivation� for�Active�Power�Control� in�Wind�Turbines�
Active� power� control� (APC)� is� the� purposeful� control� of�the�real�power�output�of�a�wind�turbine�or�collection�of�wind�turbines�in�order�to�assist�in�balancing�total�power�generated�on�the�grid�with�total�power�consumed.�Real�(also�known�as�“active”)� power� is� the� portion� of� energy� flow� which� results�in� a� net� transfer� of� energy;� this� is� in� contrast� to� imaginary�or� reactive� power� flow,� which� reverses� direction� over� each�cycle� with� no� net� energy� transfer.� While� reactive� power�may� be� controlled� through� the� turbine’s� power� electronics�(even�when�the�turbine�is�offline),�real�power�is�regulated�by�changing�the�actual�amount�of�power�delivered�to�the�utility�grid.� In� doing� so,� APC� may� allow� wind� power� to� provide�important� ancillary� services� to� the� electrical� grid� and� assist�in�maintaining�an�acceptable� frequency.�
Intrinsically,� wind� turbines� do� not� provide� APC� services�and�they�have�not�historically�been�required�to�provide�such�responses� [21].� Most� modern� turbine� generators� are� decou-pled� from�grid� frequency� through�power� electronics� (type�3�or� 4),� as� described� in� Section� II-B.� Therefore,� the� inertia�of� the� generator� and� the� turbine� rotor� do� not� automatically�participate� in� the� grid� inertial� response� as� would� traditional�synchronous�generators.�Further,�because�of�this�decoupling,�changes� in� grid� frequency�do�not� elicit� an� automatic� gover-nor� response� that� is� common� with� conventional� generation�sources.�
Increasingly,�there�are�two�main�motivations�for�why�APC�should� be� provided� by� wind� turbines.� The� first� motivation�is� that� regulation� is�essential� for�maintaining�grid� frequency�and� as� wind� penetration� increases� it� can� provide� key� sup-port� in� maintaining� the� required� balance.� A� FERC/LBNL�(Federal�Energy�Regulatory�Commission/Lawrence�Berkeley�National�Laboratory)�study�discusses�a�recent�decline�in�grid�frequency�response,�and�states�that�although�increasing�wind�penetration� is� not� the� cause,� frequency� response� could� be�improved� through� the� expanded� use� of� frequency� control�capabilities�in�wind�turbines�[22,�23].�Further,�a�recent�study�by� the� IPCC� (Intergovernmental� Panel� on� Climate� Change)�
found�that�although�low�to�medium�wind�penetrations�(up�to�20%�of�annual�demand)�poses�no�“insurmountable�technical�barriers,”� higher� levels� could� require� additional� flexibility�options� such� as� greater� use� of� wind� power� curtailment� and�output�control�[24].�As�wind�energy�penetration�increases,�it�is� potentially� displacing� regulation-providing� generators.� At�very� high� wind� penetration� levels� it� becomes� necessary� for�wind�turbines�to�also�provide�these�services�[21]�as�it�has�for�several�European�grid�operators.�
In�countries�and�regions�with�relatively� isolated�grids�and�relatively� large� levels� of� wind� penetration,� participation� in�grid�frequency�regulation�by�wind�turbines�and�wind�plants�is�crucial.�A�2010�report�issued�by�Project�UpWind�[2]�indicates�that� replacing� conventional� generation� sources� with� a� large�percentage�of�wind�power�not�capable�of�active�power�control�can� potentially� have� significant� impacts� on� stability� of� grid�frequency.�This�effect�is�pronounced�on�island�grids�with�low�levels�of�interconnectivity�to�other�grids,�such�as�many�of�the�Greek� isles� [25].�
The�necessity�of�participation�in�grid�frequency�regulation�by�wind�turbines�is�reflected�in�the�requirements�and�regula-tions�put�on�wind�plants�by�TSOs�in�regions�with�high�levels�of�wind�penetration�or�relatively�isolated�grids.�For�example,�the� Irish� grid� code� requires� that� wind� plants� have� active�power� curtailment� capabilities,� and� outlines� specific� active�power�generation�set-points�as�a�function�of�available�power�in�the�event�of�a�frequency�deviation�[26].�This�code�further�specifies� a� minimum� response� rate� for� individual� turbines�of� 1� %� of� rated� power� per� second.� Elsewhere,� Denmark’s�TSOs� Eltra� and� Elkraft� require� that� wind� plants� be� able�to� track� a� reserve� power� offset� and� track� reference� power�levels� generated� by� the� system� operator� [27].� In� Canada,�Hydro-Quebec�requires�that�wind�plants�rated�above�10�MW�have� the� ability� to� modify� their� active� power� output� for� at�least� 10� seconds� in� response� to� grid� frequency� deviations�greater�than�0.5�Hz� [28].�Additionally,�the�Spanish�TSO,�Red�Electrica,� mandates� that� wind� plants� respond� to� frequency�deviations� with� proportional� control� of� active� power� output�within�a� range�of�percentages�of� rated�power�[29].�
The� second� motivation� for� active� power� control� by� wind�turbines� is� the�potential� to� increase� the�profitability�of�wind�plants�by�enabling�participation�in�ancillary�service�markets.�A� recent� study� by� Kirby� demonstrates� a� potential� for� wind�plants�to�“increase�their�own�profits�by�providing�regulation”�[30].�
C.� Industry�Activity� in�Support�of�APC�
The� various� active� power� control� requirements� laid� out�by�grid�operators�have�motivated�technological�development�by� wind� turbine� manufacturers.� These� developments� have�been�made�at�both� individual� turbine�and�wind�plant� scales.�Technologies� to� provide� response� in� all� of� the� inertial,� pri-mary,�and�secondary�time�scales�have�been�developed�at� the�individual� turbine� level.�For� example,�Siemens�has�patented�a�method� for�dynamically�modifying�an� individual� turbine’s�active�power�output�in�response�to�grid�frequency�deviations�
6
[figure EPRI]
• Primary control counteracts initial frequency drop and implementedvia local droop control of turbine governors
• Secondary AGC is centralized and uses integral control to restorefrequency
3 / 22
Conventional operational strategy
Central control authority
• AGC is implemented with a central regulator
• Frequency deviation ω is measured at low-voltage network andintegrated to generate the regulator output signal p
T p = ω
• The incremental contribution of the individual generatingunits to the total generation is obtained via participationsfactors K−1
i
ui = −K−1i p, i = 1, 2, . . . , n
4 / 22
Conventional operational strategy
• The conventional strategy is developed for conventionalgenerators which have high inertia, hence abrupt changes arebetter absorbed by the system, thus easing the task offrequency restoration
• Renewable generation leads to significant reduction of inertia,hence to a more volatile network, which challenges existingcontrol schemes
5 / 22
Distributed control
An answer to this challenge has leveraged the use of localcontrollers that cooperate over a communication network
ICT for Network Systems
PHYS
ICAL
NETW
ORK
COMMUN
ICATION
NETW
ORK
!1 = "1M1"1 = !A1"1 + u1 ! Pl1 + Pe1
y1 = "1
...
!n = "nMn"n = !An"n + un ! Pln + Pen
yn = "n
!1 =!
j!N1(!j ! !1) ! q"1
1 "1u1 = q"1
1 "1
...
!n =!
j!Nn(!j ! !n) ! q"1
n "nun = q"1
n "n
Pe1
Pen
!
!
"1
u1
"n
un
15 / 32
• Semi-centralized
Tp =∑
i ωi , ui = −K−1i p
• Distributed averaging integral
Ti pi =∑
j∈N ci
(pj − pi ) + K−1i ωi
ui = −K−1i pi
Yet, due to security, robustness and economic concerns, it isdesirable to regulate the frequency without relying oncommunication
6 / 22
Power network
Lossless, network-reduced power system with n generating units
POWER NETWORK
gener. unit 6 gener. unit 7 gener. unit 8
gener. unit 1 gener. unit 2
gener. unit 5gener. unit 4
gener. unit 3
generating unit ifrequency deviation ωi
controll. power inj. uiuncontroll. power inj. P?i
generating unit jfrequency deviation ωj
controll. power inj. ujuncontroll. power inj. P?j
power
transmission
[figure Stegink]
7 / 22
Frequency dynamics
θi = ωi , Mi ωi = −Aiωi −∑
j∈Ni
γij︷ ︸︸ ︷ViVjBij sin(θi − θj) + ui − P?i
Local measurements: ωi
• Swing equations
• ωi frequency deviation
• θi phase angle deviation
• voltages Vi constant
• purely inductive linesBij = Bji
• mechanical equivalent ⇒Energy function:H = −1
2
∑i 6=j
BijViVj cos(θi − θj) +1
2
∑i
Miω2i
M1 u1
ω1
M2
u2
ω2
M3
u3ω3
M4
u4
ω4
M5
u5
ω5
M6
u6 ω6M7
u7
ω7
M8
u8
ω8
γ12
γ23
γ34
γ45
γ56
γ67
γ81
γ36
[figure Stegink]8 / 22
Synchronization frequency control
θi = ωi
Mi ωi = −Diωi + ui + P?i −∑
j BijEiEj sin(θi − θj)
• Synchronous solution ωi = ωsync
• Synchronous frequency ωsync =∑
i P?i +
∑i ui∑
i Di
Case n = 2
M1ω1 = −D1ω1 + u1 + P?1 − B12E1E2 sin(θ1 − θ2)
M2ω2 = −D2ω2 + u2 + P?2 − B21E2E1 sin(θ2 − θ1)
If ω1 = ω2 = ωsync = const, summing up
0 = −(D1 + D2)ωsync + u1 + u2 + P?1 + P?
2
• Zero frequency deviation 0 =∑
i P?i +
∑i ui
9 / 22
Optimal frequency restoration
Manifold choices of u?i to achieve
0 =∑
i
P?i +∑
i
u?i
Optimal dispatch problem
minimizeu∈Rn
∑i aiu
2i
subject to∑
i P∗i +
∑i ui = 0
Solution
u?i = −a−1i
∑i P
?i∑
i ai
Fair proportional sharing
aiu?i = aju
?j ∀i , j
Optimal frequency restorationGiven unknown P?i , design
ui (ωi )
that stabilizes the powersystem model to
(θ?i , ω?i = 0, u?i )
10 / 22
Fully decentralized frequency control
Ti pi = ωi
ui = −pi
• No communicationrequired
• Frequency regulationω → 0
IEEE 39-node ‘New England’ benchmark network
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
• Frequency at G1
• Noisy measurementsωi + ηi
• Non-zero mean noise η
• Noise bound η = 0.01Hz
11 / 22
Fully decentralized frequency control
• No optimalityu → u(p(0), θ(0)) 6= u?
Active power output of all generators (no noise)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
Active power output of all generators (noise)
• Unstable behavior
• Steady state
0 ≈ Ti pi = ωi + ηi 6=0 ≈ Tj pj = ωj + ηj
12 / 22
Leaky integral control
Ti pi = ωi−Kipiui = −pi
• No communicationrequired
• Synchronousfrequency
ωsync =
∑i P
?i∑
i Di +∑
i K−1i
• Banded frequencyregulation
∑i
K−1
i ≥∑
i P?i
ε−∑i
Di ⇒ |ωsync| ≤ ε
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6, G9, G10, Ki = 0.01
for the others
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
Decentralized pure integral control13 / 22
Leaky integral control
Ti pi = ωi−Kipiui = −pi
• Steady-state
u?i = −K−1i ωsync
• Power sharing
Kiu∗i = Kju
∗j
• Approx steady-state optimality
minimizeu∈Rn∑
i Kiu2i
subject to∑
i P∗i +
∑i (1 + DiKi )ui = 0
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6,
G9, G10, Ki = 0.01 for the others
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
Decentralized pure integral control
14 / 22
Leaky integral control
Ti pi = ωi + ηi−Kipiui = −pi
• Noisy measurements ωi + ηi
• Non-zero mean noise η
• Noise bound η = 0.01Hz
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
Leaky integral control Ti = 0.05s, Ki = 0.005 for G3, G5, G6,
G9, G10, Ki = 0.01 for the others
Robust frequency regulation (ISS)
‖x(t)‖2 ≤ λe−αt‖x(0)‖2 + γ( supt∈R≥0
‖η(t)‖)2
where λ, α, γ are positive constants and
x = col(δ − δ?, ω − ω?, p − p?)
measures the deviation from the synchronous solution15 / 22
Impact of control parameters
Tuning of the gains Ki
Ki = k for G3 G5 G6 G9 G10Ki = 2k for othersTi = τ = 0.05s
As k ↗• Noise-free steady-state frequency
error ↗• α↗ implies convergence time ↘• γ ↘ implies RMSE ↘
0 0.002 0.004 0.006 0.008 0.010
0.1
0.2
Fre
qu
ency
err
or
(Hz)
0 0.002 0.004 0.006 0.008 0.0110
20
30
40
50
Co
nv
erg
ence
tim
e (s
)
0 0.002 0.004 0.006 0.008 0.01
Inverse DC gain k
3
4
5
6
7
Fre
qu
ency
RM
SE
(H
z)
×10-3
Increasing gains Ki leads to
• reduced accuracy in frequency regulation• faster response• increased robustness to noise
16 / 22
Impact of control parameters
Tuning of the time constants Ti
Ki = 0.005 for G3 G5 G6 G9 G10Ki = 0.01 for the othersTi = τ
As τ ↗• α↘ implies convergence time ↗• γ ↘ implies RMSE ↘
0 0.02 0.04 0.06 0.08 0.1
10
20
30
Conver
gen
ce t
ime
(s)
0 0.02 0.04 0.06 0.08 0.1
Time constant τ (s)
4
6
8
10
Fre
quen
cy R
MS
E (
Hz)
×10-3
Increasing time constants Ti leads to
• slower response
• increased robustness to noise
17 / 22
Tuning recommendations
0 0.002 0.004 0.006 0.008 0.010
0.1
0.2
Fre
qu
ency
err
or
(Hz)
0 0.002 0.004 0.006 0.008 0.0110
20
30
40
50
Co
nv
erg
ence
tim
e (s
)
0 0.002 0.004 0.006 0.008 0.01
Inverse DC gain k
3
4
5
6
7
Fre
qu
ency
RM
SE
(H
z)
×10-3
0 0.02 0.04 0.06 0.08 0.1
10
20
30
Co
nv
erg
ence
tim
e (s
)
0 0.02 0.04 0.06 0.08 0.1
Time constant τ (s)
4
6
8
10
Fre
qu
ency
RM
SE
(H
z)
×10-3
• Fix ratios between K−1i from
generating units operationcosts
u?i = −K−1i ωsync
• Fix∑
i K−1i for banded
frequency regulation
∑
i
K−1i ≥
∑i P
?i
ε−∑
i
Di
• Fix Ti to strike a trade-offbetween frequency rejectionrate and noise rejection
Ti ↗ ⇒ α↘ and γ ↘18 / 22
A further comparison
Decentralized inte-
gralDecentralized leaky DAI
Frequency
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
0 20 40 60 80
Time (s)
59.6
59.7
59.8
59.9
60
Fre
quen
cy (
Hz)
no noise
noisy
Power
no noise
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
Power
with noise
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
0 20 40 60 80
Time (s)
-50
0
50
100
150
Pow
er
(MW
)
DAI Ti pi =∑
j∈N ci
(pj − pi ) + K−1i ωi ui = −K−1
i pi 19 / 22
Robust stability
Proof is Lyapunov-based, using a strict Lyapunov function
W = U(δ)− U(δ∗)−∇U(δ∗)>(δ − δ∗)
+1
2(ω − ω∗)>M(ω − ω∗) +
1
2(p − p∗)>T (p − p∗)
+ε(∇U(δ)−∇U(δ∗))>M(ω − ω∗).
• U(δ) = −12
∑i 6=j BijViVj cos(δi − δj) potential energy
• W with ε = 0 is the “shifted” energy function H + 12p>Tp
• For sufficiently small ε, W is strictly decreasing along thesolutions
• This allows for quantification of robustness to noise
20 / 22
Conclusions
A fully decentralized stabilizing integral control for achieving robustnoise-rejection, satisfactory steady-state regulation, desirable transientperformance
• These objectives are not aligned and trade-offs must be found
• Tuning guidelines are provided
• Resulting time constants Ti/Ki compatible with actuator response time
• Low-pass filter compares favourably wrt droop (noise rejection)
Future work
• Lead compensators could improve transient performance
• Extension more accurate physical models
• Impact of topology on the diffusion of noise and scalability
Reference
Weitenberg, Jiang, Zhao, Mallada, De Persis, Dorfler (2018). Robust decentralized secondary frequency control inpower systems: merits and trade-offs. IEEE Transactions on Automatic Control, in press, available asarXiv:1711.07332•Weitenberg, De Persis, Monshizadeh (2018). Exponential convergence under distributed averaging integralfrequency control. Automatica, 98, 103-113.?Weitenberg, De Persis (2018). Robustness to noise of distributed averaging integral controllers. Systems &Control Letters, 119, 1-7.
21 / 22
Thank you!
22 / 22